Issue 44

V. Reut et alii, Frattura ed Integrità Strutturale, 44 (2018) 82-93; DOI: 10.3221/IGF-ESIS.44.07

The transcendental equation for (19) is equal to the transcendental equation for the first singular integral equation in (13), and has the following form

2

2

8 12 3       3 4





2

4 

  









cos

0

(20)

3 4 3 4  



The Eq. (20) is equal to the transcendental equation in [32].

A CKNOWLEDGMENTS

T

he authors are grateful to Simon Dyke for the editing of the manuscript’s text.

R EFERENCES

[1] Vorovich, I. I. and Kopasenko, V. V., (1966). Some problems of elasticity theory for the semi-strip. (in Russian), Prikladnaya matematica i mekchanica, 30(1), pp. 128-136. [2] Babeshko, V. A., Babeshko, O. M. and Evdokimova, O. V., (2010). On the method of block element, Mechanics of Solids, 45, pp. 437-444. DOI: 10.3103/S0025654410030143. [3] Duduchava, R. V., (1979). Integral equations with fixed singularities, BG Teubner. [4] Antipov, Y. A., (2015). Singular integral equations with two fixed singularities and applications to fractured composites, The Quarterly Journal of Mechanics and Applied Mathematics, 68(4), pp. 461-501. [5] Onischuk, O. V., Popov, G. Ya. and Farshayt, P. G., (1988). The problem about bend of rectangular plate with linear pile, which goes on the fixed side by one end, Mechanics of solids, 6, pp. 160-167. [6] Soldatov, A. P., (1973). A problem in function theory, Diff. Equations, 9(2), pp. 248–253. [7] Bueekner, H. F., (1960). Some stress singularities and their computation by means of integral equations, Boundary problems in differential equation, Univ. Wisconsin Press. Madison, pp. 215-230. [8] Bueekner, H. F., (1966). On a class of singular integral equations, J. Mathem. Anal. and Appl., 14, pp. 392-426. [9] Bierman, G. I., (1971). A particular class of singular integral equations, J. Appl. Mathem., 20(1), pp. 99-109. [10] Gohberg, I. and Krein, M.G., (1960). Introduction of the Theory of Linear nonselfadjoint operators, American mathematical society. Transl, 14, pp. 217-287. [11] Noble, B., (1958). Methods Based on the Wiener-Hopf Technique For the Solution of Partial Differential Equations, Belfast, Northern Ireland, Pergamon Press. [12] Tricomi, F., (1932). Atti Accad. Naz. Lincei, Ser. 5(14), pp. 134–247. [13] Michlin, S. G. and Prössdorf, S., (1986). Singular Integral Operators, Akademie Verlag, Berlin. [14] Wu X.-F., Lilla, E. and Zou, W.-S., (2002). A semi-infinite internal crack between two bonded dissimilar elastic strips, Archive of Applied Mechanics, 72, pp. 630-636. [15] Duduchava, R., Krupnik, N. and Shargorodsky, E., (1999). An algebra of integral operators with fixed singularities in kernels, Integral Equations and Operator Theory, 33(4), pp. 406-425. [16] Junghanns, P. and Rathsfeld, A., (2002). On polynomial collocation for Cauchy singular integral equations with fixed singularities, Integral Equations and Operator Theory, 43(2), pp. 155-176. [17] Savruk, M. P., Osiv, P. N. and Prokopchuk, I. V., (1989). Numerical analysis in plane problems of the crack’s theory (in Russian), Naukova dumka, Kyiv. [18] Popov, V. G., (2012). A dynamic contact problem which reduces to a singular integral equation with two fixed singularities, Journal of Applied Mathematics and Mechanics, 76(3), pp. 348-357. [19] Shabozov, M. Sh., (1995). An approach to the investigation of optimal quadrature formulas for singular integrals with fixed singularity, Ukrainian Mathematical Journal, 47(9), pp. 1479-1485. [20] Sahakyan, A. V., (2011). Method of discrete singularities for solution of singular integral and integro-differential equations, Proceedings of A. Razmadze Mathematical Institute, 156.

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